BME 130-Numerical Methods in Biomedical Engineering Lab 5-Linear Systems and Gaussian Elimination 9/19/2018 Fall 2018 DUE: 9/26/2018 LAB 5: LINEAR SYSTEMS & GAUSSIAN ELIMINATION In lecture, we have been discussing systems of linear equations. Recall that a linear equation is one that can be written in the form: Where the [x,x.. xnis aset of unknown variables, and [aa b) are a (given) set of constant coefficients. Therefore, when solving this equation for a known set of constants, we need to find the combination of numbers, that when assigned to [x,,x2, .. ). satisfies the relationship. Expanding on this definition, we often find engineering applications where a set of variables, (x,,Xz. ....X), mst satisfy more than one linear relationship. Therefore, we need multiple equations to describe our system. If we have ' number of linear relationships to solve, and have n unknown variables, then we expand our previous equation into a system of simultaneous equations: Now our solution for (x2has to satisfy all 'm' equations simultaneously. This setup is called a "system of lnear equations", and the entire field of Linear Algebra has to do with ways in which we can represent and manipulate these systems to make solving them easier. Chief among these representations are matrix representations. Recalling that our original set of coefficients (4,a--,an,b) has now been expanded to a set of coefficients for each equation, we can rewrite it as: d12 bi b2 a2n and amz Note that, for convenience, we have separated the left-side coefficients ay from the right side coefficients by. Using this notation, we can read aas "the jth coefficient inthe ith equation." Therefore, the first subscript always denotes the equation number (1 i minthis case) and the second always represents which variable the coefficient belongs to (1 SSn). Recalling our definition of matrix multiplication, our system of equations can be written a21 22a2m Or equivalently: Where A is our matrix of coefficients, and and b are our unknown and solution vectors, respectively The matrix A is matrix-multiplied by the vector to produce the vector b. If we know the values for A and , then b is just the matrix product of the two. However, more often, we know A and b, and need to find the unknown to satisfy the relationship (hence "linear algebra"). BME 130 M. Leineweber Page 1 of 10